U.S. patent application number 16/739526 was filed with the patent office on 2020-09-10 for quantum signature method and quantum secret sharing method using quantum trapdoor one-way function.
This patent application is currently assigned to KOREA INSTITUTE OF SCIENCE AND TECHNOLOGY. The applicant listed for this patent is KOREA INSTITUTE OF SCIENCE AND TECHNOLOGY. Invention is credited to Young Wook CHO, Ji Woong CHOI, Sang Wook HAN, Min Sung KANG, Yong Su KIM, Sang Yun LEE, Sung Wook MOON.
Application Number | 20200287714 16/739526 |
Document ID | / |
Family ID | 1000004606427 |
Filed Date | 2020-09-10 |
View All Diagrams
United States Patent
Application |
20200287714 |
Kind Code |
A1 |
HAN; Sang Wook ; et
al. |
September 10, 2020 |
QUANTUM SIGNATURE METHOD AND QUANTUM SECRET SHARING METHOD USING
QUANTUM TRAPDOOR ONE-WAY FUNCTION
Abstract
This specification discloses a quantum public-key cryptosystem.
The quantum public-key cryptosystem may use two rotation operators
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) satisfying a cyclic evolution. The two rotation
operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) do not have a commutation
relation or an anti-commutation relation with each other. The
commutation relation or the anti-commutation relation is
established when either of the following conditions is satisfied:
.theta.=2i.pi., .phi.=2j.pi., or {circumflex over (n)}{circumflex
over (m)}=1 (i, j=integer), and .theta.=(2k+1).pi.,
.phi.=(2l+1).pi., or {circumflex over (n)}{circumflex over (m)}=0
(k, l=integer).
Inventors: |
HAN; Sang Wook; (Seoul,
KR) ; MOON; Sung Wook; (Seoul, KR) ; KIM; Yong
Su; (Seoul, KR) ; LEE; Sang Yun; (Seoul,
KR) ; CHO; Young Wook; (Seoul, KR) ; KANG; Min
Sung; (Seoul, KR) ; CHOI; Ji Woong; (Seoul,
KR) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
KOREA INSTITUTE OF SCIENCE AND TECHNOLOGY |
Seoul |
|
KR |
|
|
Assignee: |
KOREA INSTITUTE OF SCIENCE AND
TECHNOLOGY
Seoul
KR
|
Family ID: |
1000004606427 |
Appl. No.: |
16/739526 |
Filed: |
January 10, 2020 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H04L 9/0852 20130101;
H04L 9/0861 20130101; H04L 9/0825 20130101 |
International
Class: |
H04L 9/08 20060101
H04L009/08 |
Foreign Application Data
Date |
Code |
Application Number |
Mar 6, 2019 |
KR |
10-2019-0025896 |
Claims
1. A quantum public-key cryptosystem using two rotation operators)
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) satisfying a cyclic evolution, wherein R n ^ ( .theta.
) = cos ( .theta. 2 ) I - i sin ( .theta. 2 ) ( n x .sigma. x + n y
.sigma. y + n z .sigma. z ) and ##EQU00008## R m ^ ( .PHI. ) = cos
( .PHI. 2 ) I - i sin ( .PHI. 2 ) ( m x .sigma. x + m y .sigma. y +
m z .sigma. z ) ##EQU00008.2## where {circumflex over
(n)}=(n.sub.x,n.sub.y,n.sub.z) and {circumflex over
(m)}=(m.sub.x,m.sub.y,m.sub.z) and are rotation axes, and .theta.
and .phi. are rotation angles.
2. The quantum public-key cryptosystem of claim 1, wherein the two
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfy neither of the following
conditions: .theta.=2i.pi., .phi.=2j.pi., or {circumflex over
(n)}{circumflex over (m)}=1 (i,j=integer), and .theta.=(2k+1).pi.,
.phi.=(2l+1).pi., or {circumflex over (n)}{circumflex over (m)}=0
(k,l=integer).
3. The quantum public-key cryptosystem of claim 1, comprising: a
unitary transformation configured to encrypt a message in a quantum
state |.psi.; and trapdoor information T configured to decrypt an
encrypted quantum state |.psi., wherein the unitary transformation
equals R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.), and the trapdoor
information T equals R.sub.{circumflex over
(n)}.sup..dagger.(.theta.).
4. A quantum signature method comprising: (a) generating, by a
transmitter, a quantum message |M; (b) generating, by the
transmitter, a private key and a public key by using two rotation
operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfying a cyclic evolution;
(c) generating, by the transmitter, a quantum signature |S by
applying the rotation operators R.sub.{circumflex over
(n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.); and (d)
transmitting, by the transmitter, a quantum signature pair |M|S
including the quantum message |M and the quantum signature |S to a
receiver.
5. The quantum signature method of claim 4, wherein the two
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfy neither of the following
conditions: .theta.=2i.pi., .phi.=2j.pi., or {circumflex over
(n)}{circumflex over (m)}=1 (i,j=integer), and .theta.=(2k+1).pi.,
.phi.=(2l+1).pi., or {circumflex over (n)}{circumflex over (m)}=0
(k,l=integer).
6. The quantum signature method of claim 4, wherein operation (b)
comprises: (b-1) generating, by the transmitter, an arbitrary
quantum state |.psi. and generating a private key R.sub.{circumflex
over (n)}(.theta.) satisfying |M=R.sub.{circumflex over
(n)}(.theta.)|.psi.; and (b-2) generating, by the transmitter, a
public key R.sub.{circumflex over (m)}(.phi.) satisfying a cyclic
evolution.
7. The quantum signature method of claim 6, wherein operation (b)
further comprises (b-3) transmitting, by the transmitter, the
public key R.sub.{circumflex over (m)}(.phi.) through a public
channel.
8. The quantum signature method of claim 4, wherein the quantum
signature |S is an equation below: |S=R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi..
9. The quantum signature method of claim 4, further comprising (e)
verifying, by the receiver, the quantum signature |S by applying a
public key R.sub.{circumflex over (m)}.sup..dagger.(.phi.) to the
quantum signature pair |M|S received from the transmitter as shown
in an equation below: |M|R{circumflex over
(m)}.sup..dagger.(.phi.)|S|=|.psi.|R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.|=|.psi.|e.sup.i.epsilon.|.psi.|=|e.sup.i.epsilon..paral-
lel..psi.|.psi.|=1
10. A method of sharing a quantum secret, the method comprising:
(a) generating, by a secret generator, a quantum secret |S; (b)
generating, by the secret generator, an encryption key and a
decryption key by using two rotation operators R.sub.{circumflex
over (n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.)
satisfying a cyclic evolution; (c) generating, by the secret
generator, an encrypted quantum secret |S' from the quantum secret
by using the encryption key; (d) dividing, by the secret generator,
the decryption key into N pieces and transmitting the N pieces of
decryption key to a plurality of secret receivers; and (e)
transmitting, by the secret generator, the encrypted quantum secret
|S'' to a secret verifier.
11. The method of claim 10, wherein the two rotation operators
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) satisfy neither of the following conditions:
.theta.=2i.pi., .phi.=2j.pi., or {circumflex over (n)}{circumflex
over (m)}=1 (i,j=integer), and .theta.=(2k+1).pi.,
.phi.=(2l+1).pi., or {circumflex over (n)}{circumflex over (m)}=0
(k,l=integer).
12. The method of claim 10, wherein the encryption key is
R.sub.{circumflex over (m)}.sup..dagger.(.phi.)R.sub.{circumflex
over (n)}(.theta.)R.sub.{circumflex over (m)}(.phi.), the
decryption key is R.sub.{circumflex over
(n)}.sup..dagger.(.theta.), and the encrypted quantum secret is
|S'=R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|S.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application claims priority to and the benefit of
Korean Patent Application No. 2019-0025896, filed on Mar. 6, 2019,
the disclosure of which is incorporated herein by reference in its
entirety.
BACKGROUND
1. Field of the Invention
[0002] The present disclosure relates to a quantum public-key
cryptosystem, a quantum signature method, and a method of sharing a
quantum secret and more particularly, to a quantum public-key
system employing a quantum trapdoor one-way function.
2. Discussion of Related Art
[0003] The security of modern cryptography is seriously threatened
by development of a quantum computer. In particular, public key
cryptography based on computational complexity, such as factoring
and discrete logarithm, is easily analyzed by Shor's algorithm
implemented by a quantum computer. As an alternative of such modern
cryptography, quantum cryptography was proposed. Since quantum
cryptography is based on basic principles, such as the no-cloning
theorem, the uncertainty principle, and the irreversibility of
quantum measurement, security is fundamentally ensured even in a
quantum computing environment.
[0004] Quantum Key Distribution (QKD) is a representative quantum
cryptography protocol. The QKD protocol enables communicators to
share a secret key without meeting together in person. In terms of
cryptology, the QKD protocol is symmetric key cryptography and
provides only confidentiality among characteristics of a
cryptography service, such as confidentiality, integrity,
authentication, and non-repudiation. To overcome this limitation,
an arbitrated quantum signature scheme, a quantum digital signature
scheme, etc. were proposed. However, these protocols are also
applications of the QKD protocol and thus cannot be fundamental
solutions. Consequently, quantum public-key cryptosystem is
required for quantum cryptography to provide confidentiality,
integrity, authentication, and non-repudiation.
[0005] In order to implement a quantum public-key cryptosystem, it
is necessary to develop a quantum trapdoor one-way function
first.
SUMMARY OF THE INVENTION
[0006] The present disclosure is directed to providing a quantum
public-key cryptosystem.
[0007] Objects of the present disclosure are not limited to the
aforementioned object, and other objects which have not been
mentioned will be clearly understood by those of ordinary skill in
the art from the following descriptions.
[0008] According to an exemplary embodiment of the present
disclosure, a quantum public-key cryptosystem uses two rotation
operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfying a cyclic
evolution.
R n ^ ( .theta. ) = cos ( .theta. 2 ) I - i sin ( .theta. 2 ) ( n x
.sigma. x + n y .sigma. y + n z .sigma. z ) and ##EQU00001## R m ^
( .theta. ) = cos ( .PHI. 2 ) I - i sin ( .PHI. 2 ) ( m x .sigma. x
+ m y .sigma. y + m z .sigma. z ) ##EQU00001.2##
[0009] where {circumflex over (n)}=(n.sub.x,n.sub.y,n.sub.z) and
{circumflex over (m)}=(m.sub.x,m.sub.y,m.sub.z) are rotation axes,
and
[0010] .theta. and .phi. are rotation angles.
[0011] According to an exemplary embodiment of the present
disclosure, the two rotation operators R.sub.{circumflex over
(n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.) may satisfy
neither of the following conditions:
.theta.=2i.pi., .phi.=2j.pi., or {circumflex over (n)}{circumflex
over (m)}=1 (i,j=integer), and
.theta.=(2k+1).pi., .phi.=(2l+1).pi., or {circumflex over
(n)}{circumflex over (m)}=0 (k,l=integer).
[0012] According to an exemplary embodiment of the present
disclosure, the quantum public-key cryptosystem may include: a
unitary transformation configured to encrypt a message in a quantum
state |.psi. and trapdoor information T configured to decrypt an
encrypted quantum state |.psi.'. The unitary transformation may
equal R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.), and the trapdoor
information T may equal R.sub.{circumflex over
(n)}.sup..dagger.(.theta.).
[0013] According to an exemplary embodiment of the present
disclosure, a quantum signature method includes: (a) generating, by
a transmitter, a quantum message |M; (b) generating, by the
transmitter, a private key and a public key by using two rotation
operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfying a cyclic evolution;
(c) generating, by the transmitter, a quantum signature |S by
applying the rotation operators R.sub.{circumflex over
(n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.); and (d)
transmitting, by the transmitter, a quantum signature pair |M|S
including the quantum message |M and the quantum signature |S to a
receiver.
[0014] According to an exemplary embodiment of the present
disclosure, operation (b) may include: (b-1) generating, by the
transmitter, an arbitrary quantum state |.psi. and generating a
private key R.sub.{circumflex over (n)}(.theta.) satisfying
|M=R.sub.{circumflex over (n)}(.theta.)|.psi.; and (b-2)
generating, by the transmitter, a public key R.sub.{circumflex over
(m)}(.phi.) satisfying a cyclic evolution.
[0015] According to an exemplary embodiment of the present
disclosure, operation (b) may further include (b-3) transmitting,
by the transmitter, the public key R.sub.{circumflex over
(m)}(.phi.) through a public channel.
[0016] According to an exemplary embodiment of the present
disclosure, the quantum signature |S may be an equation below:
|S=R.sub.{circumflex over (n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.
[0017] According to an exemplary embodiment of the present
disclosure, the quantum signature method may further include (e)
verifying, by the receiver, the quantum signature |S by applying a
public key R.sub.{circumflex over (m)}.sup..dagger.(.phi.) to the
quantum signature pair |M|S received from the transmitter as shown
in an equation below:
|M|R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)|S|=|.psi.|R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.|=|.psi.|e.sup.i.epsilon.|.psi.|=|e.sup.i.epsilon.||.psi-
.|.psi.|=1
[0018] According to an exemplary embodiment of the present
disclosure, a method of sharing a quantum secret includes: (a)
generating, by a secret generator, a quantum secret |S; (b)
generating, by the secret generator, an encryption key and a
decryption key by using two rotation operators R.sub.{circumflex
over (n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.)
satisfying a cyclic evolution; (c) generating, by the secret
generator, an encrypted quantum secret |S from the quantum secret
by using the encryption key; (d) dividing, by the secret generator,
the decryption key into N pieces and transmitting the N pieces of
decryption key to a plurality of secret receivers; and (e)
transmitting, by the secret generator, the encrypted quantum secret
|S' to a secret verifier.
[0019] According to an exemplary embodiment of the present
disclosure, the encryption key may be R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.), the decryption key
may be R.sub.{circumflex over (n)}.sup..dagger.(.theta.), and the
encrypted quantum secret may be |S'=R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|S.
[0020] Other details of the present disclosure are included in the
detailed description and drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0021] The above and other objects, features and advantages of the
present disclosure will become more apparent to those of ordinary
skill in the art by describing in detail exemplary embodiments
thereof with reference to the accompanying drawings, in which:
[0022] FIG. 1 is a conceptual diagram schematically showing a
cyclic evolution under a non-(anti-)commutation relation condition
and a non-cyclic evolution under a (anti) commutation relation
condition in the Special Unitary Group of Degree 2 (SU(2));
[0023] FIG. 2 is an exemplary diagram showing a cyclic evolution of
an arbitrary quantum state |.psi. realized by rotation operators
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) on the Bloch sphere;
[0024] FIGS. 3 to 6 are exemplary diagrams separately showing
cyclic evolutions of an arbitrary quantum state |.psi. realized by
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy a non-commutation
relation condition or a non-anti-commutation relation condition, on
the Bloch sphere;
[0025] FIG. 7 is a set of exemplary diagrams showing non-cyclic
evolutions of an arbitrary quantum state |.psi. realized by
rotation operators and R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy a commutation
relation condition, on the Bloch sphere;
[0026] FIG. 8 is a set of exemplary diagrams showing non-cyclic
evolutions of an arbitrary quantum state |.psi. realized by
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy an
anti-commutation relation condition, on the Bloch sphere;
[0027] FIGS. 9 to 11 show an average Uhlmann's fidelity
|.psi.|R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi.| of
R.sub.{circumflex over (n)}.sup..dagger.(.theta.)R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi. and
|.psi.;
[0028] FIG. 12 is a flowchart of a quantum signature method
according to an exemplary embodiment of the present disclosure;
[0029] FIG. 13 is a conceptual diagram of a quantum signature
method according to an exemplary embodiment of the present
disclosure;
[0030] FIG. 14 is a flowchart of a method of sharing a quantum
secret according to an exemplary embodiment of the present
disclosure; and
[0031] FIG. 15 is a conceptual diagram of a method of sharing a
quantum secret according to an exemplary embodiment of the present
disclosure.
DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS
[0032] Advantages and features of the present disclosure and
methods for achieving them will be made clear from embodiments
described below with reference to the accompanying drawings.
However, the present disclosure may be embodied in many different
forms and should not be construed as being limited to the
embodiments set forth herein. Rather, these embodiments are
provided so that this disclosure will be thorough and complete and
will fully convey the scope of the present disclosure to those of
ordinary skill in the technical field to which the present
disclosure pertains. The present disclosure is only defined by the
claims.
[0033] This specification discloses a quantum signature method and
a quantum secret sharing method employing a quantum trapdoor
one-way function. A quantum trapdoor one-way function uses a cyclic
evolution of an arbitrary quantum state and uses a condition under
which such a cyclic evolution occurs as trapdoor information. The
trapdoor information of a quantum trapdoor one-way function may be
used as a public key of a quantum public-key system, and in this
way, the trapdoor information may be used in a quantum signature
method and a quantum secret sharing method. In this specification,
it will be disclosed that that a quantum trapdoor one-way function
may be designed by using a cyclic evolution of an arbitrary quantum
state, and a quantum signature method and a quantum secret sharing
method, which are application techniques of a quantum public-key
cryptosystem based on the fact, will be described.
[0034] As for a trapdoor one-way function, like a one-way function,
it is easy to calculate an output y when an input x is given. In
reverse, however, when the output y is given, it is difficult to
calculate the input x. Only when special information referred to as
a trapdoor is given, it is possible to easily calculate the input
x. It is known that a quantum trapdoor one-way function having the
same function as the trapdoor one-way function does not exist
substantially in quantum cryptography. However, this specification
will disclose that it is possible to design a quantum trapdoor
one-way function by using a cyclic evolution of an arbitrary
quantum state.
[0035] In quantum mechanics, a cyclic evolution refers to a case in
which an arbitrary quantum state undergoes a unitary transformation
and returns to itself not via the same route. Due to such a cyclic
evolution, a global phase e.sup.i.epsilon. of an arbitrary quantum
state
.psi. = cos ( .alpha. 2 ) 0 + e i .beta. sin ( .alpha. 2 ) 1
##EQU00002##
is obtained as follows.
|.psi.=e.sup.i.epsilon.|.psi. [Equation 1]
[0036] Here, is a unitary operator, and the global phase
e.sup.i.epsilon. is not .+-.1. The cyclic evolution of Equation 1
is represented as Equation 2 and Equation 3 below by using two
single qubit unitary operators.
R n ^ ( .theta. ) = cos ( .theta. 2 ) I - i sin ( .theta. 2 ) ( n x
.sigma. x + n y .sigma. y + n z .sigma. z ) [ Equation 2 ] R m ^ (
.theta. ) = cos ( .PHI. 2 ) I - i sin ( .PHI. 2 ) ( m x .sigma. x +
m y .sigma. y + m z .sigma. z ) [ Equation 3 ] ##EQU00003##
[0037] Four evolutions may be represented as Equation 4 below by
using Equation 2 and Equation 3.
R.sub.{circumflex over (n)}.sup..dagger.(.theta.)R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.=e.sup.i.epsilon.|.psi. [Equation 4]
[0038] Here, {circumflex over (n)}=(n.sub.x,n.sub.y,n.sub.z) and
{circumflex over (m)}=(m.sub.x,m.sub.y,m.sub.z) are rotation axes,
and .theta. and .phi. are rotation angles
[0039] R.sub.{circumflex over (n)}(.theta.) of Equation 2 and
R.sub.{circumflex over (m)}(.phi.) of Equation 3 are not in a
commutation relation or an anti-commutation relation so that
Equation 4 may become a cyclic evolution. A condition under which
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) have a commutation relation [R.sub.{circumflex over
(n)}(.theta.),R.sub.{circumflex over (m)}(.phi.)]=R.sub.{circumflex
over (n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)-R.sub.{circumflex over (m)}(.phi.)R.sub.{circumflex
over (n)}(.theta.)=0 or an anti-commutation relation
[R.sub.{circumflex over (n)}(.theta.),R.sub.{circumflex over
(m)}(.phi.)]=R.sub.{circumflex over (n)}(.theta.)R.sub.{circumflex
over (m)}(.phi.)-R.sub.{circumflex over
(m)}(.phi.)R.sub.{circumflex over (n)}(.theta.)=0 is shown in
Equation 5 or Equation 6 below.
.theta.=2i.pi., .phi.=2j.pi., or {circumflex over (n)}{circumflex
over (m)}=1 (i,j=integer) [Equation 5]
.theta.=(2k+1).pi., .phi.=(2l+1).pi., or {circumflex over
(n)}{circumflex over (m)}=0 (k,l=integer) [Equation 6]
[0040] When {circumflex over (n)}{circumflex over (m)}=1 is
satisfied, {circumflex over (n)}=(n.sub.x,n.sub.y,n.sub.z) and
{circumflex over (m)}=(m.sub.x,m.sub.y,m.sub.z) are equal to each
other. When {circumflex over (n)}{circumflex over (m)}=0 is
satisfied, {circumflex over (n)}=(n.sub.x,n.sub.y,n.sub.z) and
{circumflex over (m)}=(m.sub.x,m.sub.y,m.sub.z) are orthogonal to
each other.
[0041] When R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.) satisfy the commutation relation
of Equation 5 and the anti-commutation relation of Equation 6, a
global phase e.sup.i.alpha. in Equation 4 becomes .+-.1 as shown in
Equation 7 below.
R{circumflex over (n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.=.+-.R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(m)}(.phi.)|.psi.=1|.psi. [Equation 7]
[0042] FIG. 1 is a conceptual diagram schematically showing a
cyclic evolution under a non-(anti) commutation relation condition
and a non-cyclic evolution under a (anti) commutation relation
condition in the Special Unitary Group of Degree 2 (SU(2)).
[0043] FIG. 2 is an exemplary diagram showing a cyclic evolution of
an arbitrary quantum state |.psi. realized by rotation operators
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) on the Bloch sphere.
[0044] Referring to FIG. 2, the arbitrary quantum state |.psi.
initially starts at a location {circle around (1)} and moves to
locations {circle around (2)}, {circle around (3)}, {circle around
(4)}, and {circle around (5)} in sequence. It is possible to see
that the final location {circle around (5)} is identical to the
initial location {circle around (1)}.
[0045] FIGS. 3 to 6 are exemplary diagrams separately showing
cyclic evolutions of the arbitrary quantum state |.psi. realized by
the rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy a non-commutation
relation condition or a non-anti-commutation relation condition, on
the Bloch sphere.
[0046] Referring to FIG. 3, the arbitrary quantum state |.psi.
rotates counterclockwise from the initial location {circle around
(1)} to the location {circle around (2)} by the rotation operator
R.sub.{circumflex over (m)}(.phi.).
[0047] Referring to FIG. 4, the quantum state at the location
{circle around (2)} rotates counterclockwise to the location
{circle around (3)} by the rotation operator R.sub.{circumflex over
(n)}(.theta.).
[0048] Referring to FIG. 5, the quantum state at the location
{circle around (3)} rotates to the location {circle around (4)} by
a rotation operator R.sub.{circumflex over
(m)}.sup..dagger.(.phi.). The rotation operator R.sub.{circumflex
over (m)}.sup..dagger.(.phi.) is an inverse rotation operator of
the rotation operator R.sub.{circumflex over (m)}(.phi.) and
rotates clockwise about the same rotation axis.
[0049] Referring to FIG. 6, the quantum state at the location
{circle around (4)} rotates to the location {circle around (5)} by
a rotation operator R.sub.{circumflex over
(n)}.sup..dagger.(.theta.). The rotation operator R.sub.{circumflex
over (n)}.sup..dagger.(.theta.) is an inverse rotation operator of
the rotation operator R.sub.{circumflex over (n)}(.theta.) and
rotates clockwise about the same rotation axis.
[0050] FIGS. 3 to 6 show an example in which an arbitrary quantum
state rotates counterclockwise first and then inversely rotates
clockwise. However, this is merely an example, and rotation
directions are not limited to the example shown in the
drawings.
[0051] FIG. 7 is a set of exemplary diagrams showing non-cyclic
evolutions of an arbitrary quantum state |.psi. realized by the
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy a commutation
relation condition, on the Bloch sphere.
[0052] FIG. 8 is a set of exemplary diagrams showing non-cyclic
evolutions of an arbitrary quantum state |.psi. realized by the
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.), which satisfy an
anti-commutation relation condition, on the Bloch sphere.
[0053] It is possible to see that the rotation operators
R.sub.{circumflex over (n)}(.theta.) and R.sub.{circumflex over
(m)}(.phi.) shown in FIGS. 7 and 8 satisfy a commutation relation
condition or an anti-commutation relation condition and thus the
arbitrary quantum state |.psi. returns to its original location.
Therefore, the rotation operators R.sub.{circumflex over
(n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.) satisfying a
commutation relation condition or an anti-commutation relation
condition are not suitable for use in a quantum public-key
cryptosystem.
[0054] FIGS. 9 to 11 show an average Uhlmann's fidelity
|.psi.|R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi.| of
R.sub.{circumflex over (n)}.sup..dagger.(.theta.)R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi. and
|.psi..
[0055] In FIGS. 9 to 11, R.sub.{circumflex over (m)}(.phi.) is
fixed, and a rotation axis {circumflex over (n)} and a rotation
angle .theta. of R.sub.{circumflex over (n)}(.theta.) are changed.
In FIGS. 9 to 11, |.psi. is |z.+-., |x.+-., and |y.+-.,
respectively.
[0056] Referring to FIGS. 9 to 11, a point at which a cyclic
evolution occurs varies according to |.psi. as shown by a green
point (a point indicated by a solid-line arrow). However, a point
at which a non-cyclic evolution occurs is fixed as shown by a blue
solid line (a line indicated by a broken-line arrow).
[0057] In this specification, the unitary transformation which
transforms the arbitrary quantum state |.psi. into another quantum
state |.psi.' is defined as a quantum trapdoor one-way function,
and trapdoor information is defined as T. Details thereof are as
follows: [0058] Input state:
[0058] .psi. = cos ( .alpha. 2 ) 0 + e i .beta. sin ( .alpha. 2 ) 1
##EQU00004## [0059] Output state: |.psi.'=|.psi. [0060] Unitary
transformation: =R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.), [R.sub.{circumflex
over (n)}(.theta.),R.sub.{circumflex over (m)}(.phi.)].noteq.0,
{R.sub.{circumflex over (n)}(.theta.),R.sub.{circumflex over
(m)}(.phi.)}.noteq.0 [0061] Trapdoor information:
T=R.sub.{circumflex over (n)}.sup..dagger.(.theta.)
[0062] The unitary transformation is a combination
R.sub.{circumflex over (m)}.sup..dagger.(.phi.)R.sub.{circumflex
over (n)}(.theta.)R.sub.{circumflex over (m)}(.phi.) of the
rotation operators R.sub.{circumflex over (n)}(.theta.) and
R.sub.{circumflex over (m)}(.phi.). In this case, R.sub.{circumflex
over (n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.) satisfy
[R.sub.{circumflex over (n)}(.theta.),R.sub.{circumflex over
(m)}(.phi.)].noteq.0 and {R.sub.{circumflex over
(n)}(.theta.),R.sub.{circumflex over (m)}(.phi.)}.noteq.0. When the
output state |.psi.'=|.psi. is given, it is very difficult to
obtain the input state |.psi. by inverse operation. In this case,
it is possible to easily acquire the quantum state
e.sup.i.alpha.|.psi. by applying the trapdoor information
T=R.sub.{circumflex over (n)}.sup..dagger.(.theta.) to the output
state |.psi.'=|.psi.. This is represented by Equation 8 below.
T|.psi.'=T|.psi.R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.=e.sup.i.alpha.|.psi. [Equation 8]
[0063] e.sup.i.alpha.|.psi. of Equation 8 differs from the input
state |.psi. only in the global phase e.sup.i.alpha.. When
e.sup.i.alpha.|.psi. and |.psi. are input to a swap test, the
result is given by Equation 9 below.
1/2{|0.sub.ancilla[e.sup.i.alpha.|.psi..sub.1)|.psi..sub.2+|.psi..sub.1(-
e.sup.i.alpha.|.psi..sub.2)]+|1.sub.ancilla[(e.sup.i.alpha.|.psi..sub.1)|.-
psi..sub.2-|.psi..sub.1(e.sup.i.alpha.|.psi..sub.2)]} [Equation
9]
[0064] Since Equation 9 becomes Equation 10 given below, the
measurement results of ancilla qubits become |0.sub.ancilla at all
times.
0 ancilla [ 1 2 ( e i .alpha. | .psi. 1 ) | .psi. 2 + .psi. 1 ( e -
i .alpha. | .psi. 2 ) ] [ Equation 10 ] ##EQU00005##
[0065] Consequently, it is determined in the swap test that the two
quantum states e.sup.i.alpha.|.psi. and |.psi. are identical. It
has been described above that the green point (indicated by a
solid-line arrow) of FIGS. 9 to 11 represents a point at which an
Uhlmann's fidelity |.psi.|e.sup.i.alpha.|.psi. of the quantum
states e.sup.i.alpha.|.psi. and |.psi. equals 1. Another important
characteristic of a trapdoor one-way function is that it is not
possible to know what kind of operation corresponds to a function
only from trapdoor information. A point at which a cyclic evolution
occurs varies according to |.psi. as shown by the green point
(indicated by a solid-line arrow) of FIGS. 9 to 11. Therefore,
although the trapdoor information T=R.sub.{circumflex over
(n)}.sup..dagger.(.theta.) is acquired, it is not possible to know
R.sub.{circumflex over (m)}(.phi.) when |.psi. is not acquired.
Consequently, the quantum trapdoor one-way function according to
this specification does not allow estimation of the unitary
transformation =R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.) only based on the
trapdoor information T=R.sub.{circumflex over
(n)}.sup..dagger.(.theta.) like a trapdoor one-way function of
current cryptography. Due to this characteristic of a quantum
trapdoor one-way function, when the quantum trapdoor one-way
function is applied to a cryptosystem, it is possible to ensure
security and also provide various cryptography services.
[0066] A quantum signature method and a quantum secret sharing
method of a quantum public-key cryptosystem employing a quantum
trapdoor one-way function will be described below.
[0067] A quantum signature method will be described first.
Communicators include a transmitter Alice and a receiver Bob.
[0068] FIG. 12 is a flowchart of a quantum signature method
according to an exemplary embodiment of the present disclosure.
[0069] FIG. 13 is a conceptual diagram of a quantum signature
method according to an exemplary embodiment of the present
disclosure.
[0070] Referring to FIGS. 12 and 13, in a quantum signature method
according to an exemplary embodiment of the present disclosure, the
transmitter Alice generates a quantum message |M corresponding to a
message first (Operation S10).
[0071] Subsequently, the transmitter Alice may generate a private
key and a public key using two rotation operators R.sub.{circumflex
over (n)}(.theta.) and R.sub.{circumflex over (m)}(.phi.)
satisfying a cyclic evolution (Operation S20).
[0072] More specifically, the transmitter Alice generates an
arbitrary quantum state
.psi. = cos ( .alpha. 2 ) 0 + e i .beta. sin ( .alpha. 2 ) 1
##EQU00006##
and generates a private key R.sub.{circumflex over (n)}(.theta.)
satisfying |M=R.sub.{circumflex over (n)}(.theta.)|.psi.. Also, the
transmitter Alice generates a public key R.sub.{circumflex over
(m)}(.phi.) satisfying a cyclic evolution such R.sub.{circumflex
over (n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.=e.sup.i.alpha.|.psi.. Additionally, the
transmitter Alice may transmit a generated public key
R.sub.{circumflex over (m)}.sup..dagger.(.phi.) to the receiver Bob
through a public channel.
[0073] Subsequently, the transmitter Alice may generate a quantum
signature |S=R.sub.{circumflex over (n)}(.theta.)R.sub.{circumflex
over (m)}(.phi.)|.psi. by consecutively applying the operators
R.sub.{circumflex over (m)}(.phi.) and R.sub.{circumflex over
(n)}(.theta.) to the arbitrary quantum state |.psi. (Operator
S30).
[0074] Subsequently, the transmitter Alice may transmit a quantum
signature pair |M|S including the quantum message |M corresponding
to the message and the quantum signature |S to the receiver Bob
(Operation S40).
[0075] Subsequently, to verify the quantum signature pair |M|S
received from the transmitter Alice, the receiver Bob may apply the
public key R.sub.{circumflex over (m)}.sup..dagger.(.phi.) to the
quantum signature |S as shown in Equation 11 below (Operation
S50).
R.sub.{circumflex over (m)}.sup..dagger.(.phi.)|S=R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|.psi. [Equation
11]
[0076] More specifically, as shown in Equation 12 below, the
receiver Bob may check an Uhlmann's fidelity |M|R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)|S| with respect to R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)|S of FIG. 11 to which the quantum
message |M and the public key R.sub.{circumflex over
(m)}.sup..dagger.(.phi.) are applied.
|M|R{circumflex over
(m)}.sup..dagger.(.phi.)|S|=|.psi.|R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|.psi.|=|.psi.|e.sup.i.epsilon.|.psi.|=|e.sup.i.epsilon..paral-
lel..psi.|.psi.|=1 [Equation 12]
[0077] At this time, the receiver Bob accepts the quantum signature
pair of the transmitter Alice when a value of the Uhlmann's
fidelity |M|R.sub.{circumflex over (m)}.sup..dagger.(.phi.)|S| is
1, and rejects the quantum signature otherwise. The Uhlmann's
fidelity may be implemented through a swap test.
[0078] Next, a quantum secret sharing method will be described.
Communicators include a secret generator Alice, a plurality of
secret receivers Bob #1 to Bob # N, and a secret verifier
Charlie.
[0079] FIG. 14 is a flowchart of a method of sharing a quantum
secret according to an exemplary embodiment of the present
disclosure.
[0080] FIG. 15 is a conceptual diagram of a method of sharing a
quantum secret according to an exemplary embodiment of the present
disclosure. Referring to FIGS. 14 and 15, first, the secret
generator Alice may generate a quantum secret
S = cos ( .alpha. 2 ) 0 + e i .beta. sin ( .alpha. 2 ) 1
##EQU00007##
(Operation S100).
[0081] Subsequently, the secret generator Alice may generate an
encryption key R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.) and a decryption
key R.sub.{circumflex over (n)}.sup..dagger.(.theta.) by applying
operators R.sub.{circumflex over (m)}(.phi.) and R.sub.{circumflex
over (n)}(.theta.) to the quantum secret |S (Operation S200). Here,
R.sub.{circumflex over (m)}(.phi.) and R.sub.{circumflex over
(n)}(.theta.) may satisfy a cyclic evolution such as
R.sub.{circumflex over (n)}.sup..dagger.(.theta.)R.sub.{circumflex
over (m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|S=e.sup.i.epsilon.|S
[0082] Subsequently, the secret generator Alice may generate an
encrypted quantum secret |S' from the quantum secret |S using the
encryption key R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.) (Operation S300).
More specifically, the encrypted quantum secret satisfies
|S'=R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over (m)}(.phi.)|S.
[0083] Subsequently, the secret generator Alice may divide the
decryption key R.sub.{circumflex over (n)}.sup..dagger.(.theta.)
into N pieces such as R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.1)R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.2) . . . R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.N). Then, the secret generator Alice
may transmit the divided pieces of decryption key R.sub.{circumflex
over (n)}.sup..dagger.(.theta..sub.1)R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.2) . . . R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.N) to the plurality of secret
receivers Bob #1 to Bob # N (Operation S400). In this case, the
divided pieces of decryption key may be obtained by dividing the
whole rotation angle of the decryption key with respect to the same
rotation axis.
[0084] Subsequently, the secret generator Alice may transmit the
encrypted quantum secret |S' to the secret verifier Charlie
(Operation S500). Operation S400 and Operation S500 may be
simultaneously performed or may be understood in reverse order.
[0085] A subsequent secret restoration process is as follows.
[0086] Each of the plurality of secret receivers Bob #1 to Bob # N
meets the secret verifier Charlie while carrying a divided piece of
decryption key R.sub.{circumflex over (n)}(.theta..sub.i) received
from the secret generator Alice. The plurality of secret receivers
Bob #1 to Bob # N restores the quantum secret |S by applying the
divided piece of decryption key R.sub.{circumflex over
(n)}(.theta..sub.i) to the encrypted quantum secret |S' of the
secret verifier Charlie. This is represented by Equation 13
below.
R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.1)R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.2) . . . R.sub.{circumflex over
(n)}.sup..dagger.(.theta..sub.N)|S'=R.sub.{circumflex over
(n)}.sup..dagger.(.theta.)R.sub.{circumflex over
(m)}.sup..dagger.(.phi.)R.sub.{circumflex over
(n)}(.theta.)R.sub.{circumflex over
(m)}(.phi.)|S=e.sup.i.epsilon.|S [Equation 13]
[0087] Meanwhile, in the descriptions of the quantum public-key
cryptosystem, the quantum signature method, and the quantum secret
sharing method according to exemplary embodiments of the present
disclosure, communicators are basically communication devices which
are connected to each other through a quantum channel and a public
channel. The quantum channel is a communication channel which may
transmit and receive photon signals, and the public channel is a
communication channel which may transmit and receive electrical
signals. In the quantum public-key cryptosystem, the quantum
signature method, and the quantum secret sharing method according
to exemplary embodiments of the present disclosure, a quantum state
which is difficult to store or handle is not used as trap
information, and quantum operator information rather than a quantum
state is used as trapdoor information. Consequently, it is possible
to solve problems that are difficult to solve in existing quantum
cryptosystems. The quantum operator information is transmitted and
received through the public channel.
[0088] According to an aspect of the present disclosure, a
public-key cryptosystem may be provided in a quantum communication
environment.
[0089] According to another aspect of the present disclosure, since
quantum operator information rather than a quantum state which is
difficult to store or handle is used as trap information, it is
possible to ensure quantum stability and provide the same function
as a trapdoor one-way function of current cryptography.
[0090] According to another aspect of the present disclosure, it is
possible to use a single quantum state operator rather than a
multidimensional quantum state operator, such as controlled not
(CNOT), Quantum Fourier Transform, and Grover Iteration.
[0091] Effects of the present disclosure are not limited to those
mentioned above, and other effects which have not been mentioned
will be clearly understood by those of ordinary skill in the art
from the above descriptions.
[0092] Although exemplary embodiments of the present disclosure
have been described with reference to the accompanying drawings,
those of ordinary skill in the art will appreciate that various
modifications and equivalents may be made from the exemplary
embodiments without departing from the technical spirit or
essential characteristics of the present disclosure. Therefore, the
above-described embodiments should be construed as illustrative
rather than limiting in all aspects.
* * * * *